Solve 1/x+3+1/2x-1=11/7x+9

$$\frac { 1 } { x + 3 } + \frac { 1 } { 2 x - 1 } = \frac { 11 } { 7 x + 9 }$$

Solve for x

$$x=-17$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,-\frac{9}{7},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(x+3\right)\left(7x+9\right), the least common multiple of x+3,2x-1,7x+9.

$$\left(2x-1\right)\left(7x+9\right)+\left(x+3\right)\left(7x+9\right)=\left(2x-1\right)\left(x+3\right)\times 11$$

Use the distributive property to multiply 2x-1 by 7x+9 and combine like terms.

$$14x^{2}+11x-9+\left(x+3\right)\left(7x+9\right)=\left(2x-1\right)\left(x+3\right)\times 11$$

Use the distributive property to multiply x+3 by 7x+9 and combine like terms.

$$14x^{2}+11x-9+7x^{2}+30x+27=\left(2x-1\right)\left(x+3\right)\times 11$$

Combine 14x^{2} and 7x^{2} to get 21x^{2}.

$$21x^{2}+11x-9+30x+27=\left(2x-1\right)\left(x+3\right)\times 11$$

Combine 11x and 30x to get 41x.

$$21x^{2}+41x-9+27=\left(2x-1\right)\left(x+3\right)\times 11$$

Add -9 and 27 to get 18.

$$21x^{2}+41x+18=\left(2x-1\right)\left(x+3\right)\times 11$$

Use the distributive property to multiply 2x-1 by x+3 and combine like terms.

$$21x^{2}+41x+18=\left(2x^{2}+5x-3\right)\times 11$$

Use the distributive property to multiply 2x^{2}+5x-3 by 11.

$$21x^{2}+41x+18=22x^{2}+55x-33$$

Subtract 22x^{2} from both sides.

$$21x^{2}+41x+18-22x^{2}=55x-33$$

Combine 21x^{2} and -22x^{2} to get -x^{2}.

$$-x^{2}+41x+18=55x-33$$

Subtract 55x from both sides.

$$-x^{2}+41x+18-55x=-33$$

Combine 41x and -55x to get -14x.

$$-x^{2}-14x+18=-33$$

Add 33 to both sides.

$$-x^{2}-14x+18+33=0$$

Add 18 and 33 to get 51.

$$-x^{2}-14x+51=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -14 for b, and 51 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\times 51}}{2\left(-1\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\times 51}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-14\right)±\sqrt{196+4\times 51}}{2\left(-1\right)}$$

Multiply 4 times 51.

$$x=\frac{-\left(-14\right)±\sqrt{196+204}}{2\left(-1\right)}$$

Add 196 to 204.

$$x=\frac{-\left(-14\right)±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$x=\frac{-\left(-14\right)±20}{2\left(-1\right)}$$

The opposite of -14 is 14.

$$x=\frac{14±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{14±20}{-2}$$

Now solve the equation x=\frac{14±20}{-2} when ± is plus. Add 14 to 20.

$$x=\frac{34}{-2}$$

Divide 34 by -2.

$$x=-17$$

Now solve the equation x=\frac{14±20}{-2} when ± is minus. Subtract 20 from 14.

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=-17 x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,-\frac{9}{7},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(x+3\right)\left(7x+9\right), the least common multiple of x+3,2x-1,7x+9.

$$\left(2x-1\right)\left(7x+9\right)+\left(x+3\right)\left(7x+9\right)=\left(2x-1\right)\left(x+3\right)\times 11$$

Use the distributive property to multiply 2x-1 by 7x+9 and combine like terms.

$$14x^{2}+11x-9+\left(x+3\right)\left(7x+9\right)=\left(2x-1\right)\left(x+3\right)\times 11$$

Use the distributive property to multiply x+3 by 7x+9 and combine like terms.

$$14x^{2}+11x-9+7x^{2}+30x+27=\left(2x-1\right)\left(x+3\right)\times 11$$

Combine 14x^{2} and 7x^{2} to get 21x^{2}.

$$21x^{2}+11x-9+30x+27=\left(2x-1\right)\left(x+3\right)\times 11$$

Combine 11x and 30x to get 41x.

$$21x^{2}+41x-9+27=\left(2x-1\right)\left(x+3\right)\times 11$$

Add -9 and 27 to get 18.

$$21x^{2}+41x+18=\left(2x-1\right)\left(x+3\right)\times 11$$

Use the distributive property to multiply 2x-1 by x+3 and combine like terms.

$$21x^{2}+41x+18=\left(2x^{2}+5x-3\right)\times 11$$

Use the distributive property to multiply 2x^{2}+5x-3 by 11.

$$21x^{2}+41x+18=22x^{2}+55x-33$$

Subtract 22x^{2} from both sides.

$$21x^{2}+41x+18-22x^{2}=55x-33$$

Combine 21x^{2} and -22x^{2} to get -x^{2}.

$$-x^{2}+41x+18=55x-33$$

Subtract 55x from both sides.

$$-x^{2}+41x+18-55x=-33$$

Combine 41x and -55x to get -14x.

$$-x^{2}-14x+18=-33$$

Subtract 18 from both sides.

$$-x^{2}-14x=-33-18$$

Subtract 18 from -33 to get -51.

$$-x^{2}-14x=-51$$

Divide both sides by -1.

$$\frac{-x^{2}-14x}{-1}=-\frac{51}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{14}{-1}\right)x=-\frac{51}{-1}$$

Divide -14 by -1.

$$x^{2}+14x=-\frac{51}{-1}$$

Divide -51 by -1.

$$x^{2}+14x=51$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=51+7^{2}$$

Square 7.

$$x^{2}+14x+49=51+49$$

Add 51 to 49.

$$x^{2}+14x+49=100$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{100}$$

Simplify.

$$x+7=10 x+7=-10$$

Subtract 7 from both sides of the equation.

$$x=3 x=-17$$